To determine the possible values for the third side of a triangle when two sides are given as 7 inches and 11 inches, we need to apply the triangle inequality theorem. The theorem states:
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
2. Mathematically, for sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] of a triangle:
- [tex]\(a + b > c\)[/tex]
- [tex]\(a + c > b\)[/tex]
- [tex]\(b + c > a\)[/tex]
Let [tex]\(a = 7\)[/tex], [tex]\(b = 11\)[/tex], and [tex]\(c\)[/tex] be the third side.
First, we consider:
[tex]\[ a + b > c \][/tex]
[tex]\[ 7 + 11 > c \][/tex]
[tex]\[ 18 > c \][/tex]
So, [tex]\(c\)[/tex] must be less than 18.
Next, we consider the absolute difference between the lengths of the two sides to ensure it establishes a lower boundary:
[tex]\[ |a - b| < c \][/tex]
[tex]\[ |7 - 11| < c \][/tex]
[tex]\[ 4 < c \][/tex]
So, [tex]\(c\)[/tex] must be greater than 4.
Combining both conditions, we get:
[tex]\[ 4 < c < 18 \][/tex]
Since [tex]\(c\)[/tex] is specified to be an integer, the possible integer values for [tex]\(c\)[/tex] are:
[tex]\[ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 \][/tex]
From these values, the smallest possible value for [tex]\(c\)[/tex] is 5, and the largest possible value for [tex]\(c\)[/tex] is 17.
The product of the smallest and largest values of [tex]\(c\)[/tex] is:
[tex]\[ 5 \times 17 = 85 \][/tex]
Thus, the product of the smallest and largest values that the third side could be is [tex]\( \boxed{85} \)[/tex].
